CompSci 100 8.1

Review of Recursion with Big-Oh Recursion is an indispensable tool in a

programmer’s toolkit Allows many complex problems to be solved simply Elegance and understanding in code often leads to

better programs: easier to modify, extend, verify (and sometimes more efficient!!)

Sometimes recursion isn’t appropriate, when it’s bad it can be very bad---every tool requires knowledge and experience in how to use it

The basic idea is to get help solving a problem from coworkers (clones) who work and act like you do Ask clone to solve a simpler but similar problem Use clone’s result to put together your answer

Need both concepts: call on the clone and use the result

CompSci 100 8.2

Keys to Recursion

Recursive functions have two key attributes There is a base case, sometimes called the halting or exit

case, which does not make a recursive call All other cases make a recursive call, with some parameter

or other measure that decreases or moves towards the base caseo Ensure that sequence of calls eventually reaches the

base caseo “Measure” can be tricky, but usually it’s straightforward

Most of you have seen the following before in CompSci 6 This time, go faster Concentrate on Recurrence Relations and Big-Oh Should now be able to make a more sophisticated

evaluation

CompSci 100 8.3

Print words entered, but backwards Can use a vector, store all the words and

print in reverse order The vector is probably the best approach, but recursion

works too

void printReversed() { // some I/O details omitted String word; word = console.readLine(); if (word.length() > 0) { // get something? printReversed(); // print the rest reversed

System.out.println(word); // then print the word }}

// somewhere in main ---.printReversed();

The function printReversed reads a word, prints the word only after the clones finish printing in reverse order

Each clone has its own version of the code, its own word variable

Recurrence Relations? Big-Oh? Space?

CompSci 100 8.4

Exponentiation Computing xn means multiplying n

numbers (or does it?) What’s the easiest value of n to compute xn? If you want to multiply only once, what can you

ask a clone?

/** @return x^n */double power(double x, int n){ if (n == 0){

return 1.0; } return x * power(x, n-1);}

Recurrence Relations? Big-Oh?

CompSci 100 8.5

Faster exponentiation

How many recursive calls are made to computer 21024? How many multiplies on each call? Is this better?/** @return x^n */

double power(double x, int n){

if (n == 0) {

return 1.0;

}

double semi = power(x, n/2);

if (n % 2 == 0) {// is n even?

return semi*semi;

}

return x * semi * semi; {// n must be odd

}

Recurrence Relations? Big-Oh?

CompSci 100 8.6

Classic examples of recursion

For some reason, computer science uses these examples: Factorial: we can use a loop or recursion. Is this an issue? Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, …

o F(n) = F(n-1) + F(n-2), why isn’t this enough? What’s needed?

o Classic example of bad recursion, to compute F(6), the sixth Fibonacci number, we must compute F(5) and F(4). What do we do to compute F(5)? Why is this a problem?

Towers of Hanoio N disks on one of three pegs, transfer all disks to

another peg, never put a disk on a smaller one, only on larger

o Every solution takes “forever” when N, number of disks, is large

CompSci 100 8.7

Fibonacci: Don’t do this recursively

/** @param n >= 0 * @return n-th Fibonacci * number */long recFib(int n){ if (0 == n || 1 == n) { return 1; } else { return recFib(n-1) + recFib(n-2); }} How many clones/calls to

compute F(5)? Recurrence Relations?

Big-Oh?

54

3 2

3

2

02 111 0

1

01

How many calls of F(1)?

How many total calls?

consider caching code

CompSci 100 8.8

Towers of Hanoi Move n disks from one peg to

another with restrictive rules/** disks moved from peg 'from‘ * to peg 'to' using peg 'aux' * @param numDisks on peg # from * @param from source peg # * @parm to target peg # * @param aux peg # for parking */void move(int from, int to, int aux, int numDisks) { if (numDisks == 1) { System.out.println("move " + from + " to " + to ); } else {

move(from,aux,to, numDisks - 1); move(from,to,aux, 1); move(aux,to,from, numDisks - 1);

}} Recurrence Relations? Big-

Oh?

Peg#1 #2 #3

CompSci 100 8.9

What’s better: recursion/iteration?

There’s no single answer, many factors contribute Ease of developing code assuming other factors ok Efficiency (runtime or space) can matter, but don’t

worry about efficiency unless you know you have to In some examples, like Fibonacci numbers,

recursive solution does extra work, we’d like to avoid the extra work Iterative solution is efficient The recursive inefficiency of “extra work” can be

fixed if we remember intermediate solutions: instance variables

Instance variable: maintains value over all function calls Local variables created each time function called

CompSci 100 8.10

Fixing recursive Fibonacci

/** @param n >= 0 and n <= 30 * @return the n-th Fibonacci number */long recFib(int n) {long[] mem = new long[31]; Arrays.fill(mem, 0); return recF (n, mem);}long recF(int n, long[] mem){ if (0 == n || 1 == n) return 1; else if (mem[n] != 0) return mem[n]; else { mem[n] = recF(n-1, mem) + recF(n-2, mem); return mem[n]; }} What does mem do? Why initialize to all zeros? Recurrence too complicated for our simple

methods

CompSci 100 8.11

Recursive Max

/** @param a contains a.length elements, 0 < a.length * @param first < a.length is index to start * @return maximal element of a[first..length-1] */double recMax(double[] a, int first) { if (first == a.length-1){ // last element, done return a[first]; } double maxAfter = recMax(a, first+1); if (maxAfter < a[first]) return a[first]; else return maxAfter;}

What is base case (conceptually)? We can use recMax to implement max as

follows return recMax(a,0);

Recurrence Relations? Big-Oh?

CompSci 100 8.12

Recognizing recursion:

/** a is changed */void change(int[] a, int first, int last){ if (first < last) {

int temp = a[first]; // swap a[first], a[last] a[first] = a[last]; a[last] = temp; change(a, first+1, last-1);

}}// original call (why?): change(a, 0, a.length-1);

What is base case? (no recursive calls) Recurrence Relations? Big-Oh?